Using the Smith Chart,
we will investigate some of the application and uses of the
Smith Chart. For the text, we will use my new "mini
Smith Chart" which is reproduced below. Clearly, there is not
much detail here, and our answers will not be as accurate as they
would be if we used a full size chart, but we want to get ideas
across here, not the best number possible, and with the small
size, we won't run out of paper before everything is done.
Note that we have a couple of "extras" on the chart. The two
scales at the bottom of the chart can be used to either set or
measure radial variables such as the magnitude of the reflection
coefficient
|Γ|
Γ
, or the VSWR, as it turns out that in practice, what
one can actually measure on a line is the VSWR. Remember, there
is a direct relationship between the VSWR and the magnitude of
the reflection coefficient.
VSWR=1+|Γ|1-|Γ|
VSWR
1
Γ
1
Γ
(1)
|Γ|=VSWR-1VSWR+1
Γ
VSWR
1
VSWR
1
(2)
Since
|rs|=|Γ|
r
s
Γ
, once we have the VSWR, we have
|rs|
r
s
and so we know how big a circle we need on the Smith
Chart in order to go from one place to the next. Note also that
there is a scale around the outside of the chart which is given
in fractions of a wavelength. Since
rs
r
s
rotates around at a rate
2βs
2
β
s
and
β=2πλ
β
2
λ
, we could either show distance in cm or something, and
then change the scale whenever we change wavelength. Or, we
could just use a distance scale in
λλ, and measure all
distances in units of the wavelength. This is what we shall do.
Since the rate of rotation is
2βs
2
β
s
, one trip around the Smith Chart is the same as going
one half of a wavelength down the line. Rotation in a clockwise
direction is the same as moving away from the load towards the
generator, while motion along the line in the other direction
(towards the load) calls for counterclockwise rotation. The
scale is, of course, a relative one, and so we will have to
re-set our zero, depending upon where the load etc. are really
located. This will all become clearer as we do an example.
Let's start out with the simplest thing we can,
with just a generator, line and load Figure 2. Our
task will be to find the input impedance,
Z
in
Z
in
, for the line, so that we can figure the input
voltage.
For this first problem, we are going to start out with all the
basics. In later examples, we probably will only give lengths in
wavelengths, and impedances in terms of
Z
0
Z
0
, but let's do this the whole way through.
